components of variance in brain perfusion and the design
TRANSCRIPT
Components of Variance in Brain Perfusion and the Design of Studies of Individual Differences: The
Baseline Study
Roberto Viviani,* Eun-Jin Sim,* Hanna Lo,* Sven Richter,*
Sebastian Haffer,* Nadine Osterfeld,* Jan Thöne,+ Georg Grön,*
Petra Beschoner*
* Department of Psychiatry III
University of Ulm + Universitätsklinik für Neurologie,
St. Josef Hospital, Klinik der Ruhr-Universität Bochum
Corresponding author: Dr Roberto Viviani, PhD Department of Psychiatry III University of Ulm Leimgrubenweg 12 89075 Ulm, Germany Telephone: +49 731 50061569 E-mail: [email protected]
NOTICE: this is the author’s version of a work that was accepted for publication in NeuroImage. Changes
resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other
quality control mechanisms may not be reflected in this document. Changes may have been made to this work
since it was submitted for publication. A definitive version was subsequently published in NeuroImage, [46:12-
22, 2009 May 15] DOI:10.1016/j.neuroimage.2009.01.041.
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Components of Variance in Brain Perfusion and the Design of Studies of Individual Differences: The Baseline Study
Abstract Simple baseline studies correlate average perfusion levels measured at rest with individual
variables, or contrast subject groups as in case-control studies. In this methodological work,
we summarize some formal properties of the design of these studies, and investigate the
sources of variance that characterize data acquired with the arterial spin labelling technique,
with the purpose of alerting users to the main sources of variation that determine background
variance and affect the power of statistical tests. This design typology is characterized by two
variance components: between acquisitions and between subjects. We show that variation
between acquisitions is affected by the presence of large vessels and venous sinuses, with
potential adverse effects especially in the temporal and insular regions, and provide maps of
the number of acquisitions or subjects required to reach the desired estimate precision.
Furthermore, we show that the largest source of variation between subjects is captured by
global perfusion levels, and can in principle be removed by adjusting the data. Significance
levels, however, are not always only improved by the adjustment procedure; we provide an
example in the correlation with age, and attempt to explain the consequences of the
adjustment with the help of a principal component analysis of the data. We also show the
existence of variation between subjects in the perfusion in the territory of the posterior
cerebral artery and in hemispheric asymmetry.
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Components of Variance in Brain Perfusion and the Design of Studies of Individual Differences: The Baseline Study
Introduction Arterial spin labeling (ASL) is a magnetic resonance imaging technique for the estimation of
regional blood perfusion (rCBF) in the brain (see Buxton 2002 for a comprehensive
introduction). This technique is particularly appropriate in the study of states or conditions
lasting one minute or longer (Aguirre et al. 2002). Unlike positron emission tomography
(PET), ASL allows the convenient and economical acquisition of quantitative brain imaging
data without exposing participants to ionizing radiation. Furthermore, the spatial and temporal
definition of ASL is generally superior to that of PET. However, until recently, technical
limitations prevented the acquisition of more than a small number of slices. These limitations
have been superseded by the introduction of new ASL sequences that maintain the dose of
transferred energy within acceptable limits (such as the continuous arterial spin labeling
sequence, CASL, introduced by Wang et al. 2005).
This study concerns the application of the CASL technique in the baseline condition. In
this condition, participants are placed in the scanner and are asked to lie quietly and rest
without falling asleep. Studies of baseline brain activity have a history going back to PET
studies of baseline metabolism, both in normal subjects, where the effect of individual
variables such as gender or age was investigated (see Tumeh et al. 2007, Willis et al. 2002 for
recent reviews), and in neurological and psychiatric disorders such as depression, to name one
example (Drevets 2000, Mayberg 2003). More recently, baseline cerebral metabolism has also
been shown to involve relative activation of medial and temporoparietal cortical areas
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(Shulman et al. 1997), thought to reflect the operation of a ‘default mode network’ during rest
(Raichle et al. 2001). Because of their safety and practicality, the CASL technique is a
promising methodology for the study of the issues raised by the baseline condition (see for
example Abler et al. 2008; Beschoner et al. 2008; O'Gorman et al. 2006; Rao et al. 2007).
An important issue in the design of any study is the way in which variance in the data
arising from unknown variables or experimental error may be controlled to improve the
capacity of the statistical test to detect the effect of interest. Understanding this aspect of
study design requires, on the one hand, considering the formal aspects of the model, and on
the other, knowledge about the magnitude and spatial distribution of undesired variance in the
data. The issues discussed here apply to all designs that are formally identical to the baseline
study, i.e. studies of individual variance where only one condition is measured, and there is no
within-subjects explanatory or experimental variable. In contrast to most experimental
studies, baseline studies focus on between-subjects variables such as belonging to a patient or
control group, or any other variable measured on the individuals.
The random effects model corresponding to the simple baseline design contains two
sources of variance: acquisition-to-acquisition and subject-to-subject variation. In the
following, our objective is the characterization of these variance components in the perfusion
images of a sample of 228 healthy young subjects. These two sources of variance may be of
interest to experimenters, because they give indication of the gains in the precision of
estimates obtained by increasing the duration of acquisition sessions (Snedecor and Cochran
1967). Differences in these quantities in different parts of the brain determine differences in
the power of statistical tests, and therefore of the regional sensitivity of the ASL technique. In
particular, we will be concerned with showing the impact of the presence of large vessels in
the variance of the perfusion signal, demonstrating the existence of a relationship between
macroscopic anatomy and signal variance in the images obtained with this technique.
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We will further explore the spatial characteristics of subject-to-subject variance through
component analyses of the averages images in each subject (the input to the second-level
analysis). This analysis will demonstrate the existence of components of variation related to
the distribution of vascular territories, and the differential impact of global perfusion levels on
white and grey matter. We will then show the relation between these components of variation
and individual variables such as age and gender.
A further issue about which we seek clarification is the impact of adjusting for global
cerebral perfusion values (gCBF) by adding them as a covariate in the analysis (Friston et al.
1990, Macey et al. 2004). Adjustment for gCBF is a pragmatic attempt to handle variance
associated with a random effect of volumes and its interaction with a voxel factor without
modeling this effect explicitly, which would involve abandoning the SPM modeling strategy.
This procedure is designed to remove global perfusion signal changes that, by increasing the
variance of the signal, can reduce the sensitivity of the statistical test (Arndt et al. 1996,
Gavrilescu et al. 2002). While the literature on adjusting for gCBF levels is concerned with its
impact on the within-subjects effects of experimental manipulations, we will be here
specifically concerned with its impact on the inference about between-subjects variables, and
will correspondingly attempt to identify the subject-to-subject variation removed by the
adjustment. A case study of the association between regional perfusion levels and age will
illustrate the effect of adjusting for gCBF levels in terms of its impact on separate components
of variance. We will also show that gCBF levels commonly correlate with individual
variables, so that the ensuing analyses (with and without adjustment) provide qualitatively
different indications about the association between perfusion levels and the individual
variable of interest.
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Material and methods
Recruitment Participants were recruited from local schools and university and by local announcements.
Exclusion criteria were neurological or medical conditions, use of medication, or a history of
mental illness, and subclinical structural abnormalities. The study is based on 228 right-
handed participants (101 males) aged between 17 and 52 years at the time of the scan (mean
age 24.7, std. dev. 5.4) who gave informed consent. The study protocol was approved by the
local ethical committee and was in compliance with national legislation and the Code of
Ethical Principles for Medical Research Involving Human Subjects of the World Medical
Association.
Data acquisition All magnetic resonance imaging (MRI) data were obtained with a 3-Tesla Magnetom Allegra
(Siemens, Erlangen, Germany) MRI system equipped with a head volume coil. All
participants were scanned at the Department of Psychiatry of the University of Ulm. A
standard T2-weighted structural brain scan from the clinical screening routine in use in our
hospital (TR 4120, TE 82) was taken on all participants to exclude subclinical structural
abnormalities. A continuous arterial spin-labeling technique was used as described in Wang et
al. (2005). Interleaved images with and without labeling were acquired for 8 min (120
acquisitions) by using a gradient-echo echo-planar imaging sequence with a field of view of
22 cm. Image size was 64 × 64 × 15 voxels, slice thickness 6 mm with a gap of 1.5 mm,
giving a voxel size of 3.44 × 3.44 × 7.50 mm. The images were acquired with an echo-planar
imaging sequence (EPI) with TR 4000, TE 17, anterior-to-posterior phase encoding, a flip
angle of 90°, and a bandwidth of 3005 Hz/Pixel. A delay of 1 sec was inserted between the
end of the labeling pulse and image acquisition to reduce transit artifacts. The SPM2 package
was used (Wellcome Department of Cognitive Neurology, London; online at
http://www.fil.ion.ucl.ac.uk) for realignment and stereotactic normalization to an EPI
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template (Montreal Neurological Institute, resampling size: 2 × 2 × 2 mm). Reconstruction of
rCBF values was obtained using the Perf_resconstruct_V02 SPM add-on software by H. Y.
Rao and J. J. Wang, from the Department of Radiology and Center for Functional
Neuroimaging at University of Pennsylvania (online at
http://www.cfn.upenn.edu/perfusion/software.htm). The software implements eq. (1) of Wang
et al. (2003). The ‘simple subtraction’ method was used. All volumes were smoothed using an
isotropic Gaussian kernel of full width half-maximum (FWHM) of 6 mm prior to the principal
component analysis. An explicit mask was obtained by combining an a priori thresholded
tissue probability maps provided by the SPM package at 0.25 for gray or white matter with
another mask thresholding the standard deviation of the mean images to less than 25 (as
described in the text). Furthermore, slices lower than z = −24 mm. were excluded, since very
low slices have very large variance in our data (these slices are close to where the labeling
pulse was given). We also excluded slices above z = 48 mm. to prevent lack of coverage of
the top of the brain in some individuals to influence the outcome of the principal component
analysis.
Time-of-flight (TOF) angiography images were obtained from a subset of 42
participants (13 males) aged between 18 and 52 (mean age 28, std. dev. 9). A T1-weighted
imaging sequence was used with a TR 41, TE 4.92, flip angle 90, and bandwidth 110
Hertz/Pixel. Image size was 512 × 640 × 56 voxels, slice thickness 1.8 mm with a gap of 18
mm between slices. These parameters correspond to a standard clinical TOF image, with the
exception that no saturation band was present in our sequence. This band is usually applied at
the upper part of the cerebrum to suppress signal from venous sinuses, which is present in our
images. These images were normalized to a template prepared as the mean of the original
images.
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Statistical analysis After preprocessing, standard ANOVA estimators in a one-way random effect model were
used to compute the acquisition-to-acquisition and subject-to-subject variance components.
Principal component analysis was carried out using standard methods, summarized in the
Appendix of Viviani et al. (2005). Data were single-centered voxel by voxel, thus considering
voxels to be ‘variables’ and the average individual volumes to be ‘observations’ in the usual
principal component terminology (Jolliffe 1986). The principal component analysis delivers
the spatial components (‘eigenimages’), and a principal component score, one for each
volume. This score is the inner product between the eigenimage and the rCBF values in each
voxel of the volume, and represents the extent to which each volume displays the pattern
identified by the component. The correlation of gCBF, age, and gender of each average
subject image with these the principal component scores therefore provides a summary
measure of how the spatial patterns of variation of the components may be related to
individual variables.
In the analysis of the perfusion modeled as an effect of age, volumes were averaged in
each individual to obtain an estimate of the baseline perfusion, which was entered at the
‘second level’ of analysis to model subjects as a random factor. Because of the lack of
temporal autocorrelation in the signal obtained with the simple subtraction method (Aguirre et
al. 2002, Mumford et al. 2006), the individual averaging procedure yields valid tests and
correct ANOVA estimates of variance components. When indicated, adjustment for the global
signal level was obtained by including a centered covariate in the model containing global
perfusion levels estimated as the average perfusion within the masked region. No scaling
procedures such as ‘grand mean scaling’ were applied to the data. Second level statistical
analysis was performed on these averaged volumes using a permutation method to obtain
voxel-level corrected significance values (Holmes et al. 1996, Nichols and Holmes 2001)
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robust relative to distributional assumptions and with good power (Nichols and Hayasaka
2003).
Results Figure 1 gives an idea of what these images look like, when taken from a 3T scanner (details
on the acquisition methods are in the Material and methods section). On the left, an arbitrarily
chosen single acquisition is displayed. Each of these images is reconstructed from two
original echo-planar imaging (EPI) scans, one of which taken after a labeling pulse, which
tags the blood in the neck before it is distributed to the brain. The reconstruction of regional
perfusion is accomplished on the basis of a compartment model (Wang et al. 2003). The
image of a single acquisition demonstrates the large amount of noise in these data, at least as
obtained in our laboratory. The low signal-to-noise ratio justifies relatively large voxel sizes
in the acquisition. In the centre of Figure 1, an average image computed from 8 min. of
acquisitions from the same subject. Given the constraints imposed by the labeling techniques
(two EPI scans at TR = 4 sec each), 60 rCBF images were acquired in this time interval. On
the right, the average rCBF obtained from the images acquired in this way from all 228
healthy adults in the sample of this study.
FIGURE 1 ABOUT HERE
Figure 1 shows the rCBF signal to be characterized by considerable instability at the edges
and outside the brain. In the average perfusion images of a single individual (center), this
instability is evident as a gross granularity of the signal at the brain edges. Figure 2 shows the
standard deviation of these average perfusion images at three transversal slices, confirming
the existence of high variance at the edges of the brain. This high variance at the edges is
about one order of magnitude larger than within the brain, with the exception of some small
brain structures (marked with red arrows in Figure 2), which are also characterized by high
variance. Furthermore, within the brain the variance of the perfusion signal is smaller in white
than in gray matter.
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FIGURE 2 ABOUT HERE
A possible concern regarding the high variance at the edges is that it may contaminate
the cortical signal at the surface of the brain after smoothing. In the rest of this study, we
masked the image at the edge using a threshold of a standard deviation of 25 ml dl−1 min−1,
since our interest focuses on the variance of the signal arising from within the brain.
The simple baseline study model As noted in the Introduction, the variance of Figure 2 may be decomposed into two sources:
acquisition-to-acquisition and subject-to-subject variance. The former is the variance arising
from differences between each acquisition due to unaccounted effects and experimental error,
while the latter is constituted by variation in brain perfusion among individuals. Formally, in a
sample of i = 1, 2, … a subjects, in each of which j = 1, 2, … n acquisitions were made (each
obtained from two scans, with and without labeling pulse), the data may be modeled voxel by
voxel as follows:
ij i ijy A ε= + , (1)
where yij is the preprocessed voxel signal, Ai is the perfusion signal in subject i, and εij is an
experimental error term introduced in each acquisition. The signal in each subject Ai is
modelled as a random variable with mean µ (the overall mean signal level) and variance 2Aσ
(the subject-to-subject variation), assumed to be independent of the variance of the
experimental errors 2σ (the acquisition-to-acquisition variation). Because of the simple
balanced structure of this model, 2Aσ and 2σ may be estimated from ordinary ANOVA mean
sums of squares (Figure 3). If these sources of variance are normally distributed, these
ANOVA estimates are also the restricted maximum likelihood estimates.
FIGURE 3 ABOUT HERE
As it is immediately clear from Figure 3, subject-to-subject and acquisition-to-
acquisition sources of variation are characterized by completely different spatial patterns. The
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major pattern emerging from the subject-to-subject variance estimate of Figure 3 is increased
variation in gray relative to white matter areas. Subject-to-subject variation is, therefore, an
important source of the differences in variance between white and grey matter that were
visible in Figure 2. Another noticeable feature of subject-to-subject variance is that rCBF
appears to vary the most across individuals in the posterior part of the brain (cerebellum and
posterior cortical areas, red arrows).
In contrast, the acquisition-to-acquisition variance is characterized, on the one hand, by
variation at the edges of the brain that was not removed by the mask, and on the other hand,
by areas of high variation on the midline, in the insular region, and in the lower portion of the
brain. These latter areas correspond to the small structures inside the brain that displayed
larger variance in Figure 2.
To better understand the origin of acquisition-to-acquisition variance, we acquired a
smaller additional dataset of time-of-flight (TOF) angiography images (Figure 4). In these
images, vessels are easily recognizable as brighter signal. One can see that the acquisition-to-
acquisition high variance of the small structures within the brain has its source in large
vessels. At the base of the brain, one can recognize the middle cerebral artery stemming from
the internal carotid, the posterior cerebral artery, and the posterior sagittal sinus. Close to the
midline, after normalization the middle cerebral artery runs in about the same position in all
subjects, but more laterally the arteries diverge forming a dispersion fan before bending
backwards, reflecting individual variation in their precise course. All these structures are
matched by areas of comparatively larger acquisition-to-acquisition variance. Also at higher
slices (Figure 4, centre and right), one can see that the increased acquisition-to-acquisition
variance at the midline and in the insular region corresponds to important vascular structures,
such as the insular arteries, and arteries on the midline (frontal and anterior cerebral arteries
anteriorly, the pericallosal artery, and the middle occipital and choroidal arteries posteriorly).
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Also venous sinuses, such as the posterior sagittal and the rectal sinus, lead to higher
acquisition-to acquisition variance.
FIGURE 4 ABOUT HERE
Some of the acquisition-to-acquisition variance visible in the bottom row of Figure 3,
however, does not correspond to large vascular structures. This is the case for high variance
regions at the edges of the brain, at the outer edge of the cerebellum and in the olfactory
cortex, and in the posterior portion of the midline at z = 24 mm.
Impact of different sources of variation on second-level analyses The ‘first’ and ‘second level’ approach to the estimation of the random effects model
translates into averaging acquisitions in each individual for the first level step, before entering
second level analysis. Let iA ⋅ denote this averaged subject signal. Using standard ANOVA
theory (see for example Frackowiak et al. 2003), it may be shown that
( ) 2 21Var( )i AE An
σ σ⋅ = + . (2)
This equation shows that the variance of the averaged subject images iA ⋅ stems from both the
acquisition-to-acquisition and the subject-to-subject sources of variance, even if the
acquisition-to-acquisition variance is damped by the number of acquisitions.
The relation between the two variance components 2Aσ and 2σ of equation (2) is of help
in estimating the relative effect of increasing the length of baseline acquisitions relative to
increasing the number of subjects scanned in a simple baseline study (Snedecor and Cochran
1967). By increasing the number of acquisitions n, the contribution of the acquisition-to-
acquisition variance to the variance of the average subject images iA ⋅ decreases by a factor
1/n. This is highly desirable, since the variance of the iA ⋅ ’s accruing from 2σ is assumed to
be due to experimental error and will not be explained by between-subjects variables in the
model. In the box plots on the left part of Figure 3, one can see that acquisition-to-acquisition
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variance in most of the brain is one order of magnitude larger than subject-to-subject variance.
In our data, n is 60, so that we may expect acquisition-to-acquisition variance in the average
subject images iA ⋅ to range from about 2σ = 20 to 2σ = 38 (1200 to 2300 divided by 60) in
most of the brain, compared to the 80-160 range of subject-to-subject variation (the
underlying rCBF estimates are expressed in ml dl−1 min−1). Near large vessels the acquisition-
to-acquisition variance is even higher, ranging from 2σ = 35 to 2σ = 65. The relatively large
variance contributed by acquisitions-to-acquisition variance means that it may pay off to
increase the duration of the baseline experiment for longer than the 8 min. used here
(assuming that the underlying state remains stable for these longer periods of time). An
optimal tradeoff in the sample size cannot be set a priori since it requires assigning a cost to
increasing the number of acquisitions and a cost to recruiting more subjects.
The random effects model can also be used to obtain the precision of perfusion
estimates. The expected variance of the estimated perfusion level in one voxel 1
aii
y A⋅⋅ ⋅== ∑ is
given by
( )( )
2 2
Var AE ya anσ σ
⋅⋅ = + . (3)
To obtain an approximate confidence interval for an estimate of a change of perfusion levels
yβ μ⋅⋅= − , the standardized perfusion levels estimates are modeled with the Student’s t
distribution or, if the sample from which the estimates are taken is large enough, as a standard
deviate z. Elementary sampling theory then provides the approximate number of subjects a or
the number of scans n per subject required to estimate a change in perfusion with the desired
precision (Snedecor 1946, pp. 456-458). Assuming our large-sample estimates 2ˆaσ and 2σ̂ are
good enough to approximate the variances that would be obtained, on average, in a new
sample, the minimum required number of subjects a, given the number of acquisitions n, is
given by:
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( )2 2 1 2
2
ˆ ˆaz na
σ σ
β
−+= , (4)
and the minimum n given a by:
2 2
2 2 2
ˆˆa
zna zσ
β σ=
−. (5)
If the desired precision is expressed as percentage change in perfusion levels δ, the preceding
equations are modified by replacing β with 100yδ ⋅⋅ .
Note that while it is always possible to improve the precision to any desired level by
increasing the number of subjects in the sample, this will not in general be true for an increase
of the number of acquisitions. Looking at equation (3), one can see that by changing the size
of n one cannot compress the expected variance of y⋅⋅ below 2A aσ .
In the top row Figure 5, we show maps of the number of subjects required to detect a
δ = 10% change in perfusion levels at the 5% confidence level (one-sided) in a sample of 60
acquisitions per subject, corresponding to sessions lasting 8 min. Because precision is
expressed as percentage change, the requirements on the sample are more exacting for the
white matter compartment, where absolute perfusion levels are lower and the change to detect
is smaller in absolute terms. Here, samples of 60 subjects or more are required. In the gray
matter compartment, in contrast, 10% precision is reached in samples of 30 to 40 subjects.
FIGURE 5 ABOUT HERE
In the bottom row of the same figure, we show the number of acquisitions required to
achieve the same precision in a sample of 40 subjects. One can see that there are patches of
the image in the white matter compartment where a sample of 40 cannot achieve 10%
precision, irrespective of the number of acquisitions (in white), due to large subject-by-subject
variance 2A aσ relative to the required precision. Around these patches, the required number
of acquisitions increases without bounds since the desired precision is almost entirely offset
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by the subject-by-subject variance 2A aσ . In contrast, in gray matter 30 to 40 acquisitions
suffice to reach this precision level, with the exception of the areas in the temporal lobe
located in proximity of large vessels.
Even if they both improve the precision of the estimate, it is important to note that
increases of the sample size in acquisitions and in subjects are not entirely exchangeable.
These increases involve changes in the representativeness of the sample for two dimensions of
the underlying population that are entirely distinct. When baseline studies are used to
investigate associations of perfusion levels with between-subjects variables, the systematic
effects of the between-subjects variables explain part of the subject-to-subject variance, not
part of the acquisition-to-acquisition (residual) variance (see Gelman and Hill 2007, pp. 263-
264; Raudenbush and Bryk 2006, pp. 20-21). Acquisition-to-acquisition variance, however,
causes uncertainties in the estimates of the average subject images iA ⋅ ’s accounted for by the
term 2 nσ in equation (2). Hence, increasing the number of acquisitions improves the fit by
increasing the precision of the estimate of the average subject images iA ⋅ ’s, while increasing
the number of subjects improves power by increasing the degrees of freedom of the second-
level fit of the iA ⋅ ’s on the between-subjects variable. The implication of the model is that 2Aσ
contains variance that one is trying to explain, while 2σ contains measurement error only, or
the sum of innumerable effects that can be treated as random. For this reason, in the following
we focus on the structure of subject-to-subject variance.
Spatial covariation of averaged subject images The subject-to-subject variance component estimates displayed in Figure 3 provide no
information on the way in which perfusion levels co-vary spatially across the brain. For
example, even if it is clear that high variation characterizes most of the cortex, perfusion
could be high in one cortical area and low in another in one subject, while in another subject
the relation may be reversed. The net effect is that variation is high in both areas. The
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alternative possibility is that some subjects have high signal, while other subjects low signal
in all areas at the same time. This also results in high variation in the same cortical areas, but
in each subject the signal is uniform, because it co-varies across voxels.
To distinguish between these possibilities and characterize the spatial co-variance of the
signal, we carried out a principal component analysis of the averaged subject images (Figures
6 and 7). The scree plot of the eigenvalues, displayed in Figure 6, suggests that at least two
components, explaining about 45% and 10% of the spatial variance, should be considered.
The ‘eigenimage’ of the first component is of the same sign in the whole brain, capturing
individual changes in the overall level of perfusion (Figure 7, top row). This component
correlates very strongly gCBF (r = 0.999, p < 0.001). Furthermore, one can see that overall
perfusion levels affect grey matter more than white matter, which is darker in the spatial map
of Figure 7. This finding suggests that individual increases in gCBF translate in relatively
higher perfusion values in the cortex and the basal ganglia, while the perfusion in the white
matter is increased to a lesser degree.
FIGURE 6 ABOUT HERE
The ‘eigenimage’ of the second component (Figure 7, second row from top)
demonstrates that the high intersubject variation in the posterior areas of the brain (previously
detected in Figure 3, top row) is due to these regions being decoupled by the rest in terms of
perfusion levels. The areas affected by this decoupling are those that are perfused by the
posterior cerebral artery and its collaterals: the cerebellum, large part of the thalamus and
some of the posterior striatum, the calcarine cortex and the medial part of the occipital cortex,
the posterior cingulate and the adjacent medial portions of the parietal cortex.
FIGURE 7 ABOUT HERE
Even if they explain a relatively small portion of the overall variance in the principal
component analysis, also the third and fourth components display a recognizable spatial
pattern (Figure 7, lower half). The third component captures the lateralization of perfusion.
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The fourth component contains variation with mostly the same spatial distribution as that
arising from the course of large vessels. Like the third component, the fourth component
shows also some lateralization. Especially the variance arising from the insular arteries, while
present on both sides, is anticorrelated. The variance caused by large vessels may originate in
acquisition-to-acquisition variance but, as we have seen in equation 2, the variance of the
averaged baseline images is contaminated by this source of variance.
FIGURE 8 ABOUT HERE
Figure 8 displays the histograms of the first four component scores, which are the
coefficients of the projection of the volumes on the direction in space of each component. The
larger this coefficient, the more marked is the presence of the pattern of spatial distribution
identified by each eigenimage. One can check that there are no extreme values that may
determine alone the direction of any component.
Spatial components and individual variables In studies of individual differences, between-subjects variables are correlated voxel by voxel
with the observed average subject signal iA ⋅ , explaining some of the subject-to-subject
variance 2Aσ contained in the variance of iA ⋅ . Given that the principal component analysis
revealed the existence of a spatial structure of 2Aσ , a question that arises naturally is the extent
to which these patterns of variation correspond to variation associated with individual
between-subject variables. To this end, we estimated the correlation between each component
score and gCBF, age, and gender (see Methods for details), using a permutation technique to
compute significance levels corrected for the multiple tests arising from the number of
components tested (Table 1).
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TABLE 1 ABOUT HERE
FIGURE 9 ABOUT HERE
The first component (Figure 9) shows an extremely strong correlation with gCBF, as
already mentioned; gCBF correlates with no other component. The first component also
negatively correlates with age (r = −0.17, p = 0.03). Given the strong correlation between this
first component and gCBF, this finding shows that older adults are characterized by decreased
cerebral blood flow values as a whole.
The second component, which may reflect the vascular decoupling of the posterior
cerebral artery from the rest of the brain, also correlates with age (r = 0.18, p = 0.02). The
third component correlates with gender (r = 0.29, p < 0.001), suggesting an association
between lateralization of perfusion and gender. Note that, with the exception of gCBF, all
these correlations explain too small a portion of the variance identified by these components
to conclude that they can be identified with the components themselves.
The fourth component shows little correlation with individual covariates. This may be
due to this component containing prevalently acquisition-to-acquisition variation, which
arises within subjects, while individual covariates are between-subjects variables. However,
there is a trend for a weak association with gender (r = 0.16, p = 0.05), suggesting higher
variation in males.
Impact of adjustment for global perfusion levels Because of the high association between the scores on the first component and gCBF levels,
when the principal component analysis was carried out after adjusting for gCBF, components
were obtained that were virtually identical to those of the unadjusted data, with the exception
of the first component, which had simply disappeared.
In our data one also finds that the statistical analysis is qualitatively changed by the
adjustment for gCBF. This is the case because age competes with gCBF for explaining the
variance captured by the first component (the amount of variance explained by these
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individual variables may be obtained from Table 1 by squaring the correlation coefficients,
since the correlations were computed separately). On its own, gCBF explains about 99% of
this variance, while age independently explains 4% of it. Hence, some of the variance
explained by gCBF must be shared with age. This is possible because age is itself correlated
with gCBF (r = −0.17, z = −2.50, p = 0.006). Thus, the adjustment removes the variance in the
data explained by age that loads on the first component, while leaving intact the variance in
the second component.
We illustrate here this effect in Figure 10, which shows the parametric maps of the
correlation between average subject rCBF and age, with and without adjustment for gCBF,
thresholded at a liberal level to allow appreciation of the differences in the estimated maps.
The map for the analysis without gCBF adjustment reveals that age affects perfusion
negatively across the cortex, but much more markedly in the prefrontal regions, reaching
voxel-level corrected significance in the anterior medial prefrontal cortex (supplementary
motor area, BA32, x, y, z = 2, 12, 48, t = −5.6, p < 0.001, and pregenual cingulus, BA10-32,
x, y, z = 2, 12, 48, t = −5.8, p < 0.001), in the lateral prefrontal cortex (BA44, x, y, z = 52, 20,
38, t = −6.3, p < 0.001), but also in posterior regions such as BA40 in the parietal cortex (x, y,
z = 58, −38, 48, t = −4.9, p = 0.006). There is little evidence of increases in perfusion in the
brainstem and the thalamic region (x, y, z = 6, −16, −12, t = 2.8, p = 0.72). Here and in the
territory of the posterior cerebral artery perfusion is increased, but not significantly. In
contrast, the gCBF-corrected analysis reveals a relative hyperperfusion in this whole territory,
especially marked in thalamic, subthalamic and mesencephalic regions (x, y, z = 6, −16, −12,
t = 5.4, p = 0.01; x, y, z = 14, −8, 2, t = 5.1, p = 0.02). There is still evidence of localized
hypoperfusion in the prefrontal regions, which reaches voxel-level corrected significance in
the medial prefrontal cortex (BA44, x, y, z = 52, 20, 38, t = −6.5, p < 0.001), but no evidence
of reduced perfusion in more posterior regions (all significance levels reported here are voxel-
level corrected; see the online supplementary data for the tabulation of the statistical analysis).
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The persistence of a significant correlation between age and perfusion in the prefrontal region
after adjustment suggests that the effect of age is not identical with the generic differences in
cortical perfusion levels identified in the principal component analysis, but rather that it
disproportionately affects the prefrontal cortex.
FIGURE 10 ABOUT HERE
Discussion The baseline study design investigated here is formally characterized by the absence of
within-subjects experimental variables. The primary data brought to second-level analysis for
statistical inference are not estimates of contrasts between experimental conditions at the first
level (as is commonly the case in functional imaging studies), but average subject perfusion
levels to be correlated with individual between-subjects variables of interest. This design
typically applies to studies of brain perfusion at rest. However, especially in the PET
literature, studies can also be found in which subjects were asked to perform one simple task
during all acquisitions. Because they lack within-subjects variables, the design of these
studies is formally the same as the one examined here, even if the substantive interpretation of
the results might differ.
This design typology contains two components of variance: subject-to-subject and
acquisition-to-acquisition (residual). Attempts to reduce undesired variance can therefore be
directed at either of these sources. In addition, experimenters may attempt to control variance
by adjusting for global perfusion levels. It was shown that each of these sources of variance
affect the data in different ways, and that the strategies to contain them are characterized by
distinctive advantages and shortcomings.
In our data, the acquisition-to-acquisition variance was particularly strong at the edges
of the brain, where the compartment model used to estimate perfusion may not deliver
meaningful results, but also in correspondence of anatomical structures such as large vessels
and venous sinuses. Since acquisition-to-acquisition variance negatively affects statistical
− 20 −
tests in the form of uncertainty in the estimate of the average perfusion levels brought to
second level, the power of tests for differences in perfusion in these regions, i.e. in the
anterior portion of the medial face of the temporal lobe, in the insula (especially posteriorly),
and in parts of the cerebellum, may be reduced. Furthermore, we found an area of high
variance in the hemispheric fissure arising from the numerous vessels in this region. This
large variance levels might reduce power for the medial aspect of the cerebral cortex if large
smoothing kernels are used.
The increased acquisition-to-acquisition variance near large vessels may be due to the
1 sec delay used in this study, which may be not long enough to allow all label to flow into
capillary and tissue. Further studies should investigate the effect of increasing delay in
reducing this aspect of acquisition-to-acquisition variance. Alternatively, acquisition-to-
acquisition variance may be reduced by increasing the length of the measurement session, and
this strategy appears to be especially meaningful if the regions where acquisition-to-
acquisition variance is high are of specific interest. However, in this case one must assume
that the mental state underlying perfusion levels is not changed by longer sessions.
An important shortcoming of the strategy of increasing the number of acquisitions is
that it does not always replace the effect of increasing the number of subjects. Apart from the
fact that one may be more interested in generalizing over subjects than over acquisitions, it
was shown that the precision of the estimate cannot be compressed at arbitrarily low levels
only by increasing session lengths. Our data showed that this may be a problem especially if
one is interested in estimating perfusion at relative precision levels in the white matter.
In the second part of the study, we investigated the variance of the average perfusion
volumes, which mainly contains subject-to-subject variance, and the effect of adjusting for
global perfusion levels. The principal component analysis identified the major spatial patterns
in which these images differ from each other. By far the largest source of variation in this
respect was related to global perfusion levels, and revealed that these changes affect grey
− 21 −
more than white matter. Because scores on this first component were virtually identical to
estimated gCBF levels, adjusting for gCBF led to the removal of this large component,
indicating that the potential increases in power brought about by the adjustment are
considerable.
Historically, the adjustment for gCBF through a covariate was justified by early reports
showing that the global PET signal was essentially independent from regional PET signal
changes arising from experimental manipulations (Friston et al. 1990, Ramsay et al. 1993). In
the typical baseline study, there are no within-subjects experimental effects, while the
variables of interest are between-subjects. Unfortunately, the case study of regression on age
illustrates that there is no reason to assume these between-subjects variables to be
uncorrelated with global perfusion levels, so that there may be genuine effects of interest in
the correlation between gCBF and explanatory variables that are obliterated by the adjustment
(see also Aguirre et al. 1998). Both analyses of the effect of age on rCBF, with and without
adjustment, revealed different and potentially relevant aspects of the modulation of perfusion
by age.
We also identified three further components of variation that affect baseline perfusion.
One affected the brainstem, the thalami, and posterior parts of the brain comprising the
calcarine and occipital cortex. These areas correspond to the territory of the posterior cerebral
artery, suggesting a partial decoupling across subjects in levels of perfusion of these areas.
This differential pattern of perfusion may be explained by the fact that the supply of the
posterior cerebral artery differs markedly from the supply of the other arterial vessels of the
brain. The posterior cerebral artery largely receives its supply by the vertebral arteries, and for
a smaller, variable part from the internal carotid artery through the posterior communicans
arteries. In contrast, the rest of the brain is supplied by the internal carotid only.
The effect of age in baseline metabolism or perfusion levels has been the subject of a
considerable number of studies (see the reviews of Tumeh et al. 2007, Willis et al. 2002),
− 22 −
setting the stage for subsequent neuroimaging studies of the effect of age on performance
(Rypma et al. 2006). Our data confirmed the findings reported in the literature. Our purpose
was not to add to this considerable body of knowledge, but primarily that of illustrating the
effects of adjusting for gCBF in the analysis. Findings based on the present sample, which is
composed prevalently by young subjects, are not necessarily generalizable.
In the course of the analysis of the subject-to-subject variation, we also demonstrated an
association between the degree of asymmetry of hemispheric perfusion and gender. This
finding parallels similar finding on the effect of gender in asymmetry in structural images
(Good et al. 2001, Toga and Thompson 2003, Narr et al. 2007) and in diffusion tractography
(Catani et al. 2007). With respect of the design of baseline studies, these observations suggest
that gender and age may be included as a nuisance covariate in the statistical analysis. The
significance of these correlations, however, was predicated on the large size of the sample of
this study. The amount of total variance explained by these individual variables remains
small, suggesting that including them as nuisance covariates in the statistical analysis is not
mandatory in general.
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Tables
Table 1. Correlation between component scores and individual covariates
gCBF Age Gender
Component 1 r = 0.99, z = 57.9,
p < 0.001
r = −0.17, z = −2.62,
p = 0.03
r = 0.13, z = 1.92,
p = 0.20
Component 2 r = 0.02, z =0.31,
p = 0.99
r = 0.18, z = 2.78,
p = 0.02
r = 0.01, z = 0.10,
p = 0.99
Component 3 r = −0.01, z = −0.18,
p = 0.99
r = 0.06, z = 0.89,
p = 0.84
r = 0.29, z = 4.51,
p < 0.001
Component 4 r ~ 0.00, z = 0.03,
p ~ 1.00
r = 0.04, z = 0.54,
p = 0.97
r = 0.16, z = 2.49,
p = 0.05
Explanation of symbols: z: Fisher’s z-transformation of correlation coefficient; p: two-tailed
significance level, corrected for the three repeated tests for each component.
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Figure legends
Figure 1. From left to right: a single acquisition; the average perfusion in a single individual;
and the overall average perfusion in the whole sample (Talairach coordinates: z = 0 mm).
Data are in ml dl−1 min−1. Negative CBF perfusion estimates tend to occur at the edges of the
brain as a result of noise in the acquisition process exerting its effect through the perfusion
estimates of the compartment model.
Figure 2. Standard deviation of average CBF images, computed from all 228 subjects in the
sample. These images show that variance occurs at two orders of magnitude. Large variance
(in yellow) is present at the edges of the brain and in correspondence of mall intracerebral
structures, indicated with red arrows in the figure. Most of the brain parenchyma displays
lower variance (in blue); within this area, one can easily distinguish white and grey matter.
Figure 3. On the left, box plots of voxel variance levels in subject-to-subject (top) and
acquisition-to-acquisition volumes (bottom). Outliers (values beyond 1.5 times the
interquartile range) are not shown to avoid clutter. Brain images show voxel by voxel
estimates of 2Aσ , the subject-to-subject variance (top), and 2σ , the acquisition-to-acquisition
variance (or modelled experimental error, bottom). Data were smoothed with a kernel of full-
width-half-maximum (FWHM) of 6 mm.
Figure 4. Normalized TOF angiography images, showing areas where major vessels are
located (brighter signal). The falciform structure within the brain parenchyma near the
posterior edge of the brain (dark at z = 0 and 24) is a normalization artefact.
Figure 5. Maps of required sample sizes in subjects (top row) and acquisitions (bottom row)
required to achieve a precision of 10% in perfusion estimates. See text for details.
Figure 6. Screeplot of the principal component analyses of the first ten components of the
average baseline images.
Figure 7. Principal components of the mean baseline images. The images are scaled to the
same vector length of 1, and the sign of the signal is arbitrary.
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Figure 8. Histograms of the first four component scores. The projection of volumes on the
eigenimages is unitless.
Figure 9. Plots illustrating the linear association between first component scores (in the
ordinates) and gCBF (left), age (centre), and gender (right). The boxplot on the right is drawn
at the lower and upper quartile values (boxes), with a line at the median value. The whiskers
cover the extent of the rest of the data, with the exception of outliers (values beyond 1.5 times
the interquartile range), which are marked with an ‘x’.
Figure 10. Comparison of statistical parametric maps (thresholded at p = 0.01, uncorrected)
for regression on age, adjusted for gender, without adjustment for gCBF (left), and with
adjustment at second level (right). The lack of negative association between age and perfusion
in the extreme frontopolar region, evident especially at z = −11 mm, is due to signal failure
arising from a susceptibility artefact.
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Figures
Figure 1
Figure 2
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Figure 3
Figure 4
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Figure 5
0
10
20
30
40
50Screeplot
% v
aria
nce
Figure 6
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Figure 7
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Figure 8
20 30 40 50 600.5
1
1.5
2
2.5x 10
4
gCBF20 30 40 50
0.5
1
1.5
2
2.5x 10
4
agefemales males
1
1.5
2
2.5x 10
4
gender
Figure 9
Figure 10